1. Advanced Algebra/Trigonometry
Rectangular Coordinate System, Pythagorean Theorem, Distance Formula, and Midpoint Formula
Advanced Algebra/Trigonometry

2. Do Now
A triple of positive integers (a, b, c) is called a Pythagorean Triple if it satisfies the Pythagorean Theorem 𝑎 2 + 𝑏 2 = 𝑐 2 . Determine whether each triple is a Pythagorean Triple.
(9, 12, 15)
(6, 8, 10)
(3, 4, 5)
(5, 10, 15)
(7, 24, 25)

3. Rectangular Coordinate System or Cartesian Plane
Points on the x and y-axis do NOT belong to any quadrant!

4. The Pythagorean Theorem

5. The Pythagorean Theorem Example 1
Find the length of the hypotenuse.
𝑎 2 + 𝑏 2 = 𝑐 2
= 𝑐 2 𝑐=
𝑐= = 13

6. The Pythagorean Theorem and the Distance Formula
𝑑 2 = 𝑥 2 − 𝑥 𝑦 2 − 𝑦 1 2
𝑑= 𝑥 2 − 𝑥 𝑦 2 − 𝑦 1 2
Distance Formula

7. The Distance Formula Example 1
Given the points (−1, −3) and (2, 3) find the distance between them.
𝑑= −1− −3−3 2
𝑑= − −6 2
𝑑= = 45 = 9×5 =2 5

8. The Distance Formula Example 2
Given the points (1, −2) and −3, 5 , find the distance between them.

9. The Midpoint Formula
In order to find the midpoint of a line segment with endpoints ( 𝑥 1 , 𝑦 1 ) and ( 𝑥 2 , 𝑦 2 ) we find the average of the two coordinates.
Midpoint = 𝑥 1 + 𝑥 2 2 , 𝑦 1 + 𝑦 2 2

10. The Midpoint Formula Example 1
Find the midpoint of the line segment with the given endpoints (4, −1) and 2, −7 .
𝑀= , −1−7 2
𝑀= 3, −4

11. The Midpoint Formula Example 2
The midpoint of line segment JK is M (2, 1). One endpoint is J (1, 4). Find the coordinates of endpoint K.
The coordinates of endpoint K are (3, −2).
1+ x 2 =
4+ y 1 2 =
1 + x = 4
4 + y = 2
x = 3
y = – 2

12. The Distance & Midpoint Formula
Find the distance and the midpoint of the line segment with the given endpoints (6, −1) and (−9, 8).

13. Homework
Textbook pages
# 2-12 evens, evens